Mathematical Model for Computing Maximum Power Output of a PV Solar Module and Experimental Validation

A solar simulator is used for testing the panel at various irradiance and temperature ranges. Testing is done at North Dakota State University, Fargo, USA in March 2011.


Introduction
The conversion of solar radiation into electricity has been extensively studied by many authors [2,4,5,11]. In particular, photovoltaic (PV) cells allow the energy transported by electromagnetic waves (i.e., photons) to be directly converted into electricity. Predicting the performance of PV panels is essential for design engineers. From the characteristic I-V curve of a given PV cell, three key physical quantities are defined: the short-circuit current, the open-circuit voltage, and the values of current and voltage that permit the maximum power to be obtained. These variables correspond to well-defined points in the I-V plane. The determination of these points is essential for the development of appropriate PV cell models. The nonlinear and implicit relationships that exist between them, however, necessitate using iterative numerical calculations [9]. Furthermore, most of these parameters depend on both the cell temperature and the solar irradiance; therefore, the knowledge of their behavior is crucial to correctly predict the performance of PV cells and arrays.
In terrestrial applications, solar cells are generally exposed to temperatures varying from 10°C to 50°C. The performance of a solar cell is influenced by temperature as its performance parameters, namely, open-circuit voltage (V oc ), short-circuit current (I sc ), curve factor (CF), and efficiency, are temperature dependent [7]. Different models This article is a part of a Special Issue on the International Conference on Energy Security, Global Warming and Sustainable Climate (SOLARIS 2012). based on the current and voltage (I-V ) characteristic curve of a P-N junction are used to describe the behavior of PV cells. In these models, a photocurrent is associated with the generation of electron-hole pairs, while a recombination current accounts for diffusion of electrons and holes across the junction. An ideal PV cell model considers only photocurrent and recombination current, where effects of electrical resistances are neglected [9]. It is apparent that practical applications must include the effects of both of them. In addition, the P-N junction itself is modeled as double-diode without resistance for polycrystalline silicon cells, and single-diode ones are used for amorphous silicon cells [3]. It is obvious that the application of the final model must correspond to the best compromise between simplicity and accuracy.
Many mathematical models have been developed to estimate PV efficiency, power, short-circuit current, and open-circuit voltage. An excellent review of different correlations is used to determine these quantities as a function of irradiance and cell temperature [8]. An explicit model was developed in [1] to determine currents and voltages for PV points by using manufacturers' data, where the electrical current changes linearly with irradiance and temperature, while voltages are expressed by temperature coefficients and a correlation term for the irradiance. An analytical model based on manufacturers' data was presented in [6]. The authors have explicitly expressed the PV current and power as a function of voltage by using a shading linear factor expressed for open-circuit voltage loses for irradiances from 1,000 W/m 2 to 200 W/m 2 ; I-V and P -V curves can thus be produced without calculating explicit expressions of current and voltage at key operational points. It must be pointed out that most of the aforementioned works deal with models that are essentially based on linear temperature and irradiance relationships that require parameters which are not available from manufacturers' datasheets. In the current study, a solar simulator is developed and used to test an amorphous silicon PV solar module (15 watt peak, 12 volt, amorphous silicon solar panel used for battery charging) at various irradiance levels of 400 W/m 2 , 700 W/m 2 , and 1,000 W/m 2 and operation temperatures from 40°C to 100°C.

Theoretical
The effect of operating temperature on the electrical efficiency of a PV cell/module can be traced to the temperature's influence upon the current, I, and the voltage, V , as the maximum power is given by (1) It turns out that both the open-circuit voltage and the fill factor decrease substantially with temperature (as the thermally excited electrons begin to dominate the electrical properties of the semi-conductor), while the short-circuit current increases, but only slightly for c-Si solar cells [12].

I-V characteristics and maximum power output
From the measured solar cell parameters (I sc , V oc , T ), the maximum power (P mp ) can be calculated. The maximum power of a solar cell is the point on the I-V characteristic curve, at which the product P = I × V is at its maximum value. The relation between I and V for an illuminated solar cell is given by Wagner [10] in the form of (2) and (3) as below: or expressed by V : In the above equation, I il denotes the electric current generated by illumination, I 0 is the diode current, R s is the inner resistance (0.03 Ω, commonly used value in literature), and V T is the thermal voltage. In order to employ (2) or (3), the diode current, the inner resistance, as well as the thermal voltage had to be known. V T was calculated from the measured cell temperatures (T ) by using the formula where m is the diode factor (1.6), q is the charge (1.602 × 10 −19 ), and k is the Boltzmann constant (1.38×10 −23 J K −1 ). The diode current (I 0 ) was determined from the measured open-circuit voltage (V oc ) by employing the following equation: Neglecting the resistance and with V = 0, it follows from (2) (short circuit) that I il ∼ I sc . To compute P mp , a computer program is developed using Newton-Raphson's method. The present calculations referred to P mp based on that scheme. After some rearrangements and substitutions by using (2) and (3), the following equation is obtained: I mp denotes the electric current at its maximum point.
Applying Newton-Raphson's method on the previous expression yields I mp : Equation (7) clarifies Newton-Raphson's method to calculate I mp in an iterative form. The subscript i denotes the ith iteration, and f and f are denoted in (8) and its derivative in (9), respectively. Once I mp is determined, V mp and P mp can be calculated as given in (10) and (11), respectively, as follows:

Experimental
Testing of solar panel was done in the research laboratory of the Department of Computer Sciences and Electronics, North Dakota State University, Fargo, USA, during the month of March 2011. Experimental setup consisted of a solar simulator fitted with two light sources (metal halide HID lamp of 1 kW each) hung from top of the simulator roof, facing a horizontal platform which was modified to move up and down in the vertical direction ( Figure 1) using a steel rope and pulley arrangement. An amorphous silicon solar module capable of producing 15 W (maximum power output and used as battery charger) at standard test conditions was placed centrally on the movable platform in a horizontal position. Three variable resistances (rheostats) were used to apply load on the PV module for generating I-V characteristics and power conditioning curves at selected irradiance and operation temperatures. In order to control the operation temperature of the PV module, two high-speed fans were fitted and operated from below the wired mesh platform on which the PV module was placed for selective cooling of the panel. The irradiance levels falling on the PV module were varied by altering the vertical distance between the light source and the movable platform on which the module was placed. The power, voltage, and current output were recorded at each load using WT 2030 model digital power meter. Cell temperature of the module (surface temperature beneath the glass cover) was recorded using a non-contacttype handheld infrared thermometer having an operating range of −20°C to 320°C. The thermometer was pointed toward the module from 30 cm distance at three different locations on the module, and then the average was taken. Solar radiation was measured using a digital solarimeter (SL-100, KIMO, Italy).

Solar irradiance dependence on the I-V characteristics and maximum power produced
The pronounced effect of solar radiation variation on the I-V characteristics and power conditioning curves of the amorphous silicon PV module is shown in Figures 2 and 3 Figure 3 are 13.806 W p at 1,000 W/m 2 , 9.674 W p at 700 W/m 2 (29.92% less as compared to P mp generated at 1,000 W/m 2 ), and 6.598 W p at 400 W/m 2 (31.79% further lesser as compared to P mp generated at 700 W/m 2 ), respectively.  3.2 Operation temperature dependence on the I-V characteristics and maximum power output It is known that the cell operation temperature has some bearing on the I-V characteristics and maximum power output (P mp ) of a solar cell. In order to exactly quantify the effect of temperature on P mp at the same radiation level for an amorphous silicon solar PV module, simulations were performed by varying the module operating temperature from 40°C to 100°C to generate the I-V characteristic curves as shown in Figure 4 and the maximum power output (P mp ) as shown in Figure 5. It was observed that at 1,000 W/m 2 radiation level, P mp was 12.531 W p at 60°C and 10.775 W p at 100°C (14% lower). At 700 W/m 2 radiation level, P mp was 8.606 W p at 45°C and 7.761 W p at 75°C (9.8% lower). At 400 W/m 2 radiation level, P mp was 6.068 W p at 40°C and 5.697 W p at 60°C (6.11% lower).

Validation of the proposed mathematical model
The measured and computed values of P mp at various selected irradiance levels are shown in Figure 6, which   clearly indicate that there is a complete matching of the two values within less than 5% deviation, thus proving the accuracy of the developed model.

Conclusions
(a) The maximum power produced decreases by about 30% when solar irradiance decreases from 1,000 W/m 2 to 700 W/m 2 level and further lowers by about 31% when solar irradiance decreases from 700 W/m 2 to 400 W/m 2 level.
(b) The effect of operating temperature is also studied on the P mp , which shows that at the same irradiance of 1,000 W/m 2 , P mp decreases by about 14% when the cell operation temperature is lowered from 100°C to 50°C.
(c) The developed model can accurately predict the performance of the PV solar panels.